Determine if the Vector is in the Column Space

Problem

A=[[0,1,0],[0,0,1],[1,0,0]],B=[[0],[1],[0]]

Solution

1. Identify the goal, which is to determine if the vector B is in the column space of matrix A denoted Col(A) This is true if the system A*x=B has a solution.

2. Set up the augmented matrix [A|B] to solve for x

[A|B]=[[0,1,0,0],[0,0,1,1],[1,0,0,0]]

3. Perform row operations to reach row-echelon form. Swap the first and third rows ((R_1)↔(R_3).

[[1,0,0,0],[0,0,1,1],[0,1,0,0]]

4. Continue row operations by swapping the second and third rows ((R_2)↔(R_3) to reach reduced row-echelon form.

[[1,0,0,0],[0,1,0,0],[0,0,1,1]]

5. Analyze the result. The system is consistent and has the unique solution (x_1)=0 (x_2)=0 and (x_3)=1 Since a solution exists, B is in the column space of A

Final Answer

The vector *B* is in the column space of *A.

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